Fractal Functions and Wavelet Expansions Based on Several Scaling Functions

Geronimo, Jeffrey S.; Hardin, Douglas P.; Massopust, Peter

doi:10.1006/jath.1994.1085

Cited by 536 publications

(253 citation statements)

References 15 publications

Supporting

Mentioning

251

Contrasting

Unclassified

Order By: Relevance

“…Typical examples of such scaling function vectors and multi-wavelets and their corresponding filters {P k } and {Q k } are the GHM multi-wavelets [6,5] and CL multi-wavelets [4], of dimension r = 2. We remark that the GHM and (one of the) CL scaling function vectors have polynomial reproduction order 2, and hence, their corresponding multi-wavelets have vanishing moments of order 2.…”

Section: Introductionmentioning

confidence: 99%

Balanced multi-wavelets in $\mathbb R^s$

Chui

Jiang

2004

Math. Comp.

View full text Add to dashboard Cite

Abstract. The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K − 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K ≥ 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the Kbalancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K ≥ 1.

show abstract

Section: Introductionmentioning

confidence: 99%

Balanced multi-wavelets in $\mathbb R^s$

Chui

Jiang

2004

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…shows further steps for moving edge points of frame 1 are extracted using frames 2 , , and . We use "GHM" Geronimo et al (1994), "GHM_p3" Ozkaramanli (2003) and "Alpert" Alpert (1993) multi-wavelets with repeated row (MW_rr) and approximation order (MW_ap) pre-processing to obtain the edge maps. The performance of these multi-wavelets employing the DCD technique is depicted in 1 Recall that we always need three consecutive frames due to the use of DCD.…”

Section: Multi-wavelets and Moving Object Detectionmentioning

confidence: 99%

Wavelet-based Moving Object Segmentation: From Scalar Wavelets to Dual-tree Complex Filter Banks

Baradarani¹,

JonathanWu²

2010

Pattern Recognition Recent Advances

View full text Add to dashboard Cite

“…For the general theory and more examples of multiple refinable functions, we refer the reader to [5], [6], [10], [11], [12], [15], [18].…”

Section: Applications To Multiple Refinable Functionsmentioning

confidence: 99%

The limits of refinable functions

Strang

2001

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. A function φ is refinable (φ ∈ S) if it is in the closed span of {φ(2x − k)}. This set S is not closed in L 2 (R), and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every f ∈ S \ S vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in [− π] are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.

show abstract

Fractal Functions and Wavelet Expansions Based on Several Scaling Functions

Cited by 536 publications

References 15 publications

Balanced multi-wavelets in $\mathbb R^s$

Balanced multi-wavelets in $\mathbb R^s$

Wavelet-based Moving Object Segmentation: From Scalar Wavelets to Dual-tree Complex Filter Banks

The limits of refinable functions

Contact Info

Product

Resources

About